1

Fundamentals of Fundamentals of

Analytical ChemistryAnalytical Chemistry

Chapter 6Chapter 6

Random Errors in Chemical Random Errors in Chemical

AnalysisAnalysis

HomeworkHomework

�� In some cases only certain parts of a In some cases only certain parts of a

problem are marked with an asterisk problem are marked with an asterisk –– in in

that case you are only responsible (for that case you are only responsible (for

purposes of any quiz) for those parts that purposes of any quiz) for those parts that

are marked.are marked.

�� Questions 1, 2, 5, 7 Questions 1, 2, 5, 7 –– 13, 17, 1913, 17, 19

Random ErrorRandom Error

�� Cannot eliminate!Cannot eliminate!

�� Causes are the same as systematic error.Causes are the same as systematic error.

�� Measurements Measurements ‘‘closer to the edgecloser to the edge’’

�� Many times a function of the analyst!Many times a function of the analyst!

�� Random error tends to be biRandom error tends to be bi--directionaldirectional

�� Tendency over the long term to cancelTendency over the long term to cancel

�� Gaussian CurveGaussian Curve

( )2

2

2

2

1)( π

µ

πσ

−−

=x

exP

HistogramHistogram

�� With a finite set of With a finite set of

measurements, can measurements, can

plot a histogramplot a histogram

�� Values (or ranges of Values (or ranges of

values) on xvalues) on x--axisaxis

�� Frequency on yFrequency on y--axisaxis

Population StatisticsPopulation Statistics

�� Statistics from data for which Statistics from data for which every valueevery valueis known!is known!

�� Average age of students at Cameron Average age of students at Cameron

University.University.

�� Population meanPopulation mean

�� Population standard deviation Population standard deviation

N

xi∑=µ

( )

N

xi∑ −=

2µ

σ

2

Normalized Gaussian CurveNormalized Gaussian Curve

�� LetLet

�� ‘‘NormalizedNormalized’’ Gaussian Gaussian

equationequation

�� Gives plot with Gives plot with

maximum at 0 and z maximum at 0 and z

equal to one standard equal to one standard

deviation unitdeviation unit

( )σ

µ−= ix

z

πσ 2

2

2

−

=

z

ey

Area under the curveArea under the curve

�� From calculus, we know that the integral From calculus, we know that the integral

of values from the Gaussian curve gives us of values from the Gaussian curve gives us

the area under the curvethe area under the curve

�� From From --∞∞ to to ∞∞, the integral of our normalized , the integral of our normalized

function is 1function is 1

�� Defining points Defining points ±±z, area = 0.683z, area = 0.683

�� Defining points Defining points ±±2z, area = 0.9542z, area = 0.954

�� Defining points Defining points ±±3z, area = 0.9973z, area = 0.997

Gaussian curveGaussian curve

�� This gives us the relative probability of This gives us the relative probability of

finding points defined by our value of zfinding points defined by our value of z

�� ±±z, 68.3%z, 68.3%

�� ±±2z, 95.4%2z, 95.4%

�� ±±3z, 99.7%3z, 99.7%

�� We can also determine percentage and We can also determine percentage and

calculate zcalculate z

�� 90% = 90% = ±±1.64z1.64z

�� 95% = 95% = ±±1.96z1.96z

Sample StatisticsSample Statistics

�� Unfortunately, in analytical chemistry we Unfortunately, in analytical chemistry we

dondon’’t have the luxury of knowing all of the t have the luxury of knowing all of the

valuesvalues

�� Representative sampleRepresentative sample

�� May or may not be truly May or may not be truly ‘‘representativerepresentative’’

�� Must modify our approach to calculating statistical Must modify our approach to calculating statistical

values.values.

Sample StatisticsSample Statistics

�� Sample mean Sample mean ––

�� No difference in calculation from population No difference in calculation from population meanmean

�� Sample standard deviation Sample standard deviation ––�� Measure of precision Measure of precision

�� Note the denominator is equal to NNote the denominator is equal to N--11

�� Loss of one degree of freedom.Loss of one degree of freedom.

N

xx

i∑=

( )

1

2

−

−=∑

N

xxs

i

Other Statistical ValuesOther Statistical Values

�� VarianceVariance

�� Standard deviation squaredStandard deviation squared

�� Can be population (Can be population (σσ22) or sample (s) or sample (s22))

�� Standard error of the meanStandard error of the mean

�� Gaussian distribution based on single Gaussian distribution based on single

measurementmeasurement

�� May improve reliability by multiple May improve reliability by multiple

measurements.measurements.

N

ssm =

3

Comparing s and Comparing s and σσ

�� When N>20, s is a pretty good estimator When N>20, s is a pretty good estimator

of of σσ

�� Can assume that s Can assume that s ≈≈ σσ

�� Series of measurements, with no Series of measurements, with no

individual set of measurements > 20, can individual set of measurements > 20, can

calculate a calculate a ‘‘pooledpooled’’ standard deviationstandard deviation

�� Assumption is deviation is a function of the Assumption is deviation is a function of the

method and not the operator!method and not the operator!

Pooled Standard DeviationPooled Standard Deviation

( ) ( ) ( )

Nzyx

zzyyxx

pooledNNNN

NsNsNss

−+++

−++−+−=

...

1...11222

( ) ( ) ( )

Nzyx

iii

pooledNNNN

zzyyxxs

−+++

−++−+−=∑ ∑ ∑

...

...222

Since most calculators can calculate a standard Since most calculators can calculate a standard

deviation, there is a deviation, there is a ‘‘modifiedmodified’’ version of this version of this

equation that can be usedequation that can be used

Other MeasurementsOther Measurements

�� Relative standard deviationRelative standard deviation

��

�� Coefficient of VariationCoefficient of Variation

�� ssrr x 100%x 100%

�� Other valuesOther values

�� RSD in RSD in pptppt (parts per thousand)(parts per thousand)

�� SSrr x 1000x 1000

x

ssr =

RoundingRounding

�� ‘‘RoundingRounding’’ is the process of mapping from is the process of mapping from

an infinite set to a finite set.an infinite set to a finite set.

�� Results of a calculation may be any valueResults of a calculation may be any value

�� Rounding insures the values reported have Rounding insures the values reported have

the correct number of digitsthe correct number of digits

�� Significant FiguresSignificant Figures

�� Rounding is really picking the closest Rounding is really picking the closest

allowed value to your calculated valueallowed value to your calculated value

RoundingRounding

�� First look First look –– first first unretainedunretained digitdigit

�� 00--44; lesser of the allowed values; lesser of the allowed values

�� 66--99; greater of the allowed values; greater of the allowed values

�� 55 –– must investigate furthermust investigate further

�� NonNon--terminal 5terminal 5

�� Followed by other nonFollowed by other non--zero digits (16.zero digits (16.5501487)01487)

�� ALWAYS the greater of the allowed valuesALWAYS the greater of the allowed values

�� Terminal 5Terminal 5

�� NOT followed by a nonNOT followed by a non--zero digit (16.zero digit (16.55))

�� Round such that the last retained digit is evenRound such that the last retained digit is even

Significant FiguresSignificant Figures

�� Every measurement has three partsEvery measurement has three parts�� MagnitudeMagnitude

�� UnitsUnits

�� ErrorError

�� Significant figures are used to indicate the Significant figures are used to indicate the error in a measurement or the result of a error in a measurement or the result of a calculation that uses measurements.calculation that uses measurements.�� All known digits and the first interpolated digitAll known digits and the first interpolated digit

�� ±±1 in the last decimal place1 in the last decimal place

4

What Figures are Significant?What Figures are Significant?

�� All nonAll non--zero digits are significant figures (zero digits are significant figures (sfsf))

�� ZeroZero

�� Between Between sfsf –– significant (5significant (5004)4)

�� Right of Right of dpdp andand right of right of sfsf –– significant (14.5significant (14.500))

�� 0.000.00147147

�� D/N meet both conditions D/N meet both conditions –– NOT significant!NOT significant!

�� Zeros in this case tell magnitude and not errorZeros in this case tell magnitude and not error

�� 47,47,000,000000,000 (?)(?)

What Figures are Significant?What Figures are Significant?

�� AmbiguousAmbiguous

�� Clarify with scientific notationClarify with scientific notation

�� X.YYX.YY…… * 10* 10ZZ

�� X must be digit 1X must be digit 1--9 inclusive9 inclusive

�� Y,Z Y,Z –– any digitany digit

�� Significant figures determined by mantissaSignificant figures determined by mantissa

�� 1010ZZ indicates magnitudeindicates magnitude

�� Numbers that indicate magnitude ARE NOT Numbers that indicate magnitude ARE NOT sfsf

�� Any zero in the mantissa must be significant!Any zero in the mantissa must be significant!

�� Right of Right of dpdp and right of and right of sfsf

LogarithmsLogarithms

�� To determine significant figures in To determine significant figures in

logarithms, we have to understand that logarithms, we have to understand that

logarithms are aliens!logarithms are aliens!

�� Anything before the decimal is not significantAnything before the decimal is not significant

�� Anything after the decimal is significantAnything after the decimal is significant

�� 11.825.825

�� 1414.052.052

�� 00.004.004

�� All have 3 significant figures!All have 3 significant figures!

Rules for Significant Figures in Rules for Significant Figures in

CalculationsCalculations

�� Addition and/or subtractionAddition and/or subtraction

�� Round answer to the Round answer to the ‘‘least reliableleast reliable’’ decimal decimal

place.place.

�� 14.2 + 8.41 = 22.6114.2 + 8.41 = 22.61

�� 14.2 (tenths place), 8.41 (hundredths place)14.2 (tenths place), 8.41 (hundredths place)

�� Answer becomes 22.6 (rounded)Answer becomes 22.6 (rounded)

�� 5.87 + 15.1 = ?5.87 + 15.1 = ?

Rules for Significant Figures in Rules for Significant Figures in

CalculationsCalculations

�� Multiplication and/or divisionMultiplication and/or division

�� Round answer to match the factor with the Round answer to match the factor with the

fewest number of significant figures.fewest number of significant figures.

�� 14.2 * 8.41 = 119.42214.2 * 8.41 = 119.422

�� 14.2 (3 14.2 (3 sfsf), 8.41 (3 ), 8.41 (3 sfsf))

�� Answer becomes 119 (rounded)Answer becomes 119 (rounded)

�� 5.87 * 15.1 = ?5.87 * 15.1 = ?

Rules for Significant Figures in Rules for Significant Figures in

CalculationsCalculations

�� Mixed calculationMixed calculation

�� Determine the number of Determine the number of sfsf in the same order for in the same order for

which you perform the calculation.which you perform the calculation.

�� Multiple calculationsMultiple calculations

�� Save any rounding until the end of the problemSave any rounding until the end of the problem

�� ‘‘RoundRound--offoff’’ errorerror

�� LogarithmsLogarithms

�� Significant figures need to match before and after Significant figures need to match before and after calculationcalculation

�� log 1.86 x 10log 1.86 x 1055 = 5.270= 5.270

�� 1010--4.824.82 = 1.5 x 10= 1.5 x 10--55

5

Special Situations (1)Special Situations (1)

�� MeanMean

�� Decimal places for the mean should match the Decimal places for the mean should match the

decimal places of the measurements used to decimal places of the measurements used to

calculate the mean.calculate the mean.

�� Standard DeviationStandard Deviation

�� Decimal places for the standard deviation Decimal places for the standard deviation

should match the decimal places of the should match the decimal places of the

measurements used to calculate the standard measurements used to calculate the standard

deviation.deviation.

Error PropagationError Propagation

�� Also called standard deviation propagationAlso called standard deviation propagation

�� What should the standard deviation (or error) What should the standard deviation (or error)

be when numbers with a known standard be when numbers with a known standard

deviation (or error) are manipulated?deviation (or error) are manipulated?

�� Addition/subtractionAddition/subtraction

�� Multiplication/divisionMultiplication/division

�� ExponentsExponents

�� logarithmslogarithms

Error PropagationError Propagation

�� Addition/subtractionAddition/subtraction

�� Absolute error propagatesAbsolute error propagates

�� For For yy±±ssyy = = aa±±ssaa + + bb±±ssbb -- cc±±sscc

�� Multiplication/divisionMultiplication/division

�� Relative error propagatesRelative error propagates

�� For For yy±±ssyy = = aa±±ssaa * * bb±±ssbb / / cc±±sscc

( ) ( ) 222

cbay ssss ++=

222

+

+

=

c

s

b

s

a

s

y

scbay

Error PropagationError Propagation

�� Goal is to find Goal is to find ssyy!!

�� For multiplication/division, must then multiply the For multiplication/division, must then multiply the

result of the calculation by yresult of the calculation by y

�� y * (y * (ssyy/y/y) = ) = ssyy

�� y is the result of the calculation ignoring the standard y is the result of the calculation ignoring the standard

deviation (error) terms.deviation (error) terms.

�� ExponentExponent

�� For y = aFor y = axx

�� Error doesnError doesn’’t t ‘‘cancelcancel’’

�� Must then calculate Must then calculate ssyy as aboveas above

=

a

sx

y

say

Error PropagationError Propagation

�� LogarithmsLogarithms

�� If you know If you know ssaa you can calculate you can calculate ssrr/r/r directlydirectly

�� Rearrange equation to solve for Rearrange equation to solve for ssaa if you know if you know

ssrr/r/r

�� Only error propagation that relates an absolute Only error propagation that relates an absolute

error to a relative error.error to a relative error.

�� Absolute error (alien) to relative error (real)Absolute error (alien) to relative error (real)

( ) ar sr

s*10ln=

Significant FiguresSignificant Figures

Special Situations (2)Special Situations (2)

�� Rounding results after error propagationRounding results after error propagation

�� First, round the error term (First, round the error term (ssyy) to ) to one one significant figure.significant figure.

�� Second, round the answer (y) to Second, round the answer (y) to the same the same number of decimal places as the error term.number of decimal places as the error term.